## Integer Factorization: A Ceiling Square Spread and a Math QR Code!

I am once again trudging into the deep forest of how prime numbers, discrete semiprimes, and other composite numbers distribute themselves among the positive odd integers, looking for patterns. The illustrations below are from the beginning steps, taken yesterday and today, of this stage of my math study.

The table I’ve built using Apple Numbers spreads out most of the odd positive integers not greater than 675 positioned according to their representation m=c2-r where c is the smallest integer greater than the square root of m that is odd or even as (m+1)/2 is odd or even. This is the method of deriving the Ceiling Square c that is consistent with the oddness or evenness of the larger square every representation of m as the difference of two perfect squares. Recall that an odd positive m greater than 1 has as many such representations as it has factorizations m=ab, with the greater perfect square being (a+b)/2 squared, and the lesser being (a-b)/2 squared. Remainders are congruent to 0 or 1 modulo 4, in harmony with the smaller square in all of m’s Fermat factorizations, and this table so far lists all such remainders less than 100, with c going up to 26. Some of the values less than 675 would actually appear in the row for c=27. It is a peculiar ordering of the odd positive integers in this way, but it serves the purposes of my study for the time being: spreading out the distribution of numbers according to Ceiling Square vs. remainder.

The whole table is too large to be readable on this page, so here is an image of the first few columns: